60th Annual Meeting Division of Fluid Dynamics
A multiscale approach to study the
stability of long waves in
near-parallel flows
S. Scarsoglio#, D.Tordella# and W. O. Criminale*
#
Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Torino,
Italy
* Department of Applied Mathematics, University of Washington, Seattle,
Washington, Usa
EFMC7, Manchester, September 14-18, 2008
Outline
 Physical problem
 Initial-value problem
 Multiscale analysis for the stability of long waves
 Conclusions
Physical Problem
 Flow behind a circular cylinder
steady, incompressible and viscous;
 Approximation of 2D asymptotic Navier-Stokes expansions (Belan &
Tordella, 2003), 20≤Re≤100.
Initial-value problem
 Linear, three-dimensional perturbative equations in terms of vorticity and
velocity (Criminale & Drazin, 1990);
 Base flow parametric in x and Re
U(y; x0, Re)
Laplace-Fourier transform in x and z directions for perturbation
quantities:
αr
= k cos(Φ) wavenumber in x-direction
γ
Φ
= tan-1(γ/αr ) angle of obliquity
k = (αr2 + γ2)1/2 polar wavenumber
αi
= k sin(Φ) wavenumber in z-direction
0 spatial damping rate
 Periodic initial conditions for
symmetric
asymmetric
and
 Velocity field vanishing in the free stream.
Early transient and asymptotic behaviour
of perturbations
The growth function G is the normalized kinetic energy density
and measures the growth of the perturbation energy at time t.
 The temporal growth rate r (Lasseigne et al., 1999) and the angular
frequency ω (Whitham, 1974)
perturbation phase
Exploratory analysis of the transient dynamics
(c): Wave
(a):
R=100,
spatial
y0=0,
evolution
x0=6.50,
in
αthe
x direction
n0=1, forasymmetric,
k=αr=0.5,
i=0.01,
αΦi =3/8
= 0,0.01,0.05,0.1.
π, k=0.5,1,1.5,2,2.5.
(b): R=50, y0=0, x0=7,
k=0.5, Φ=0, asymmetric,
n0=1, αi = 0,0.01,0.05,0.1.
(d): R=100, y0=0, x0=11.50, k=0.7, asymmetric, αi=0.02, n0=1,
(e): R=100 x0=12, k=1.2, αi=0.01,
symmetric, n0=1, Φ= π/2, y0=0,2,4,6.
Φ=0, π/2.
(f): R=50 x0=14, k=0.9, αi=0.15,
asymmetric, y0=0, Φ= π/2, n0=1,3,5,7.
……..
(a)-(b)-(c)-(d): R=100, y0=0, k=0.6, αi=0.02, n0=1,
Φ=π/4, x0=11 and 50, symmetric and asymmetric.
>>
(a): R=100, y0=0, x0=9, k=1.7, αi =0.05, n0=1, symmetric, Φ=π/8.
Asymptotic fate and comparison with modal analysis
 Asymptotic state: the temporal growth rate r asymptotes to a constant
value (dr/dt < ε ~ 10-4).
(a)-(b): Re=50, αi=0.05, Ф=0, x0=11, n0=1, y0=0. Initial-value problem
(triangles: symmetric, circles: asymmetric), normal mode analysis (black
curves), experimental data (Williamson 1989, red symbols).
Multiscale analysis for the
stability of long waves
 Different scales in the stability analysis:
 Slow scales (base flow evolution);
 Fast scales (disturbance dynamics);
 In some flow configurations, long waves can be destabilizing (for
example Blasius boundary layer and 3D cross flow boundary layer);
 In such instances the perturbation wavenumber of the unstable wave is
much less than O(1).
Small parameter is the polar wavenumber of the perturbation:
k<<1
Full linear system
base flow
(U(x,y:Re), V(x,y;Re))
Multiple scales hypothesis
Regular perturbation scheme, k<<1:
 Temporal scales:
 Spatial scales:
Order O(1)
Order O(k)
Comparison with the full linear problem
(a)-(b): Re=100, k=0.01, Ф=π/4, x0=10, n0=1, y0=0. Full linear problem
(black circles: symmetric, black triangles: asymmetric), multiscale O(1) (red
circles: symmetric, red triangles: asymmetric).
(a): R=50, y0=0, k=0.03, n0=1, x0=12, Φ=π/4, asymmetric, αi=0.04, 0.4.
->
<-
(b): R=100, y0=0, n0=1, x0=27, Φ=0,
symmetric, αi=0.2, k=0.1, 0.01, 0.001.
(c): R=100, y0=0, k=0.02, x0=13.50,
n0=1, Φ=π/2, αi=0.08, sym and asym.
(a)-(b): R=50, y0=0, k=0.04, n0=1, x0=12, Φ=π/2, asymmetric, αi=0.005,
0.01, 0.05 (multiscale O(1)), αi=0 (full problem).
Conclusions
 Synthetic perturbation hypothesis (saddle point sequence);
 Absolute instability pockets (Re=50,100) found in the
intermediate wake;
 Good agreement, in terms of frequency, with numerical
and experimental data;
 No information on the early time history of the perturbation;
 Different transient growths of energy;
 Asymptotic good agreement with modal analysis and with
experimental data (in terms of frequency and wavelength);
 Multiscaling O(1) for long waves well approximates full
linear problem.